The very fact that Henry IV, Henry V were negotiating with the Popes shows that the Church did not have absolute authority. Yes, the Treaty of Worms empowered the Church - as a political entity foremost. Its moral authority rose up only for a few other centuries.

The notion that there are lines between Church and sovereign authorities was underscored by the Canossa and Warms episodes. For Islam, this notion still seems to be lightyears away, especially regarding the Sunni doctrine of caliphate.

Where do you draw the line between "advisory, propaganda roles" and whatever you think Islamists do?

We better have a taste for distinctions here. Concrete metrics would be: claimed necessity of own superiority (over politics, other religions); free will of sovereigns; historical evolution; eagerness of "supply vs demand".

My mentioned religions coexist in China, Japan for centuries almost without particular religious wars, and are pretty cool about "foolish" rulers. Confucianism went though the Han synthesis with Taoism and Qin legalism; distrust of the (Mongol) Yuan dynasty; Neo-Confucianism, New Confucianism variations.